Reservoir simulation is used to predict the flow of fluids in underground reservoir. The fluid flow may include oil, gas and water. Such reservoir forecasting is important in reservoir management and estimating the potential recovery from a reservoir.
Reservoir simulation is well known throughout the oil industry and in the scientific literature. A good primer on the principles behind reservoir simulation is K. Aziz and A. Settari, Petroleum Reservoir Simulation, Elsevier Applied Science Publishers, London (1979). Another description of how reservoir simulation is generally performed is described in U.S. Pat. No. 6,052,520 to Watts III et al. These references are hereby incorporated by reference in their entireties.
The following are general steps taken in a conventional reservoir simulation. First a reservoir is selected for which the rock and fluid properties are to be modeled and simulated. The reservoir is modeled and discretized into a plurality of cells. Nonlinear governing equations are constructed for each cell generally in the form of finite difference equations, which are representative of properties of rocks and fluids in the reservoir. Examples of rock properties include porosity, capillary pressure and relative permeability for each phase of fluid (oil, water, gas.) Examples of fluid properties include oil viscosity, oil formation factor (B0), and pressure, temperature, and saturation in each of the cells. Nonlinear terms in these equations are linearized to arrive at a set of linear equations for each timestep of the simulation. These linear equations can then be solved to estimate solutions for unknowns such as pressure and saturation in the cells. From these values of pressure and saturation other properties can be estimated including the overall production of oil, gas and water from the reservoir in a timestep. The aforementioned steps are repeated over many such timesteps to simulate fluid flow over time in the reservoir.
The permeability of natural formations of reservoirs displays high levels of variability and complex structures of spatial heterogeneity which spans a wide range of length scales. In order to improve the reliability of petroleum reservoir simulation models, much effort has been directed towards characterizing the reservoir properties in great detail. Consequently, efficient numerical algorithms are required to tackle the resulting high resolution simulation models (with several million grid cells). Recent advances in multi-scale methods have enhanced the ability to make predictions of flow and transport in highly detailed heterogeneous reservoir models. U.S. Pat. No. 6,823,297 to Jenny et al., U.S. Patent Appln. 20050203725 to Jenny et al. and papers by Jenny, P., S. H. Lee, and H. A. Tchelepi: 2003, Multi-Scale Finite-Volume Method for Elliptic Problems in Subsurface Flow Simulation. J. Comp. Phys. 187, 47-67; and Jenny, P., S. H. Lee, and H. A. Tchelepi: 2004, Adaptive Multi-scale Finite Volume Method for Multi-Phase Flow and Transport. Multi-scale Model. Simul. 3, (1) 2004, 50-64, describe a multi-scale finite volume (MSFV) method for single phase flow and adaptive IMPES (i.e., implicit pressure, explicit saturations. The MSFV method solves flow and transport sequentially. To compute a fine-scale flow field, dual basis functions, which are solutions of local problems on a dual grid, are used to construct a global coarse scale system of equations, which can be solved to obtain global coarse grid pressures. A velocity field is reconstructed on the fine-scale using primal basis functions. In the MSFV approach, a block based Schwarz overlapping technique is used to solve hyperbolic saturation equations (i.e. the transport problem). This MSFV method has been proven to be accurate for strongly heterogeneous problems.
Black oil formulation is often used to perform simulation studies in practice. In a black oil model, three components, which are defined as fluid phases at standard (surface) conditions, are used to represent a fluid system. The hydrocarbon system is described using two pseudo components, namely oil and gas, and the third component represents water. The phase behavior of the fluid system is described using solubilities and formation volume factors.
The conservation equations of the three pseudo components are nonlinear. They are parabolic due to capillarity (three immiscible fluid phases) and compressibility (rock and fluids). However, in most settings, reservoir displacement processes are dominated by convection, so that the pressure field is nearly elliptic, while the saturation equations display nearly hyperbolic behavior.
In the MSFV method of Jenny et al., dual-grid basis functions, which are solutions to local problems on the dual grid, are used to assemble an upscaled transmissibility field on the coarse grid. Solution of this global system yields the coarse scale pressure field. A fine scale pressure field can be reconstructed using the dual-grid basis functions. Fine scale saturation equations are then solved using the reconstructed fine scale velocity field. In the saturation equation, a Schwarz overlap method is applied directly for a primal coarse cell with local boundary conditions. For more details of the calculation of the MSFV method and the use of dual-grid basis functions, see Appendix A below.
Prior multi-scale methods typically deal with single- or two-phase flow and only consider very simplified physics. They neglect capillary pressure, gravity, and dissolution. They also assume that both the porous medium and the fluids are incompressible. Accordingly, there is a need for a multi-scale finite volume method which addresses the effects of compressibility, capillary pressure and gravity.